Question

The angle of intersection of the curves xy = a² and x² − y² = 2a² is ______________ .

Options

• 0°

• 45°

• 90°

• none of these

Answer 1

90°

 

\[\text { Given }: \]

\[xy = a^2 . . . \left( 1 \right)\]

\[ x^2 - y^2 = 2 a^2 . . . \left( 2 \right)\]

\[\text { Let} \left( x_1 , y_1 \right)\text {be the point of intersection }.\]

\[\text { On differentiating (1) w.r.t. x, we get }\]

\[x\frac{dy}{dx} + y = 0\]

\[ \Rightarrow \frac{dy}{dx} = \frac{- y}{x}\]

\[ \Rightarrow m_1 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = \frac{- y_1}{x_1}\]

\[\text { On differentiating (2) w.r.t.x, we get }\]

\[2x - 2y \frac{dy}{dx} = 0\]

\[ \Rightarrow \frac{dy}{dx} = \frac{x}{y}\]

\[ \Rightarrow m_2 = \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) = \frac{x_1}{y_1}\]

\[\text { Now,} \]

\[ m_1 \times m_2 = \frac{- y_1}{x_1} \times \frac{x_1}{y_1}\]

\[ \Rightarrow m_1 \times m_2 = - 1\]

\[ \because m_1 \times m = - 1\]

\[\text { So, the angle between the curves is } 90°\]